It came down to Edwin van der Sar turning away Nicolas Anelka's effort to clinch a second Champions League victory for United manager Sir Alex Ferguson on a night when his team were second best for much of the game.
As United's players celebrated, Terry broke down in tears and was comforted by Blues coach Avram Grant.
The match was forced into penalties after the match finished at 1-1.
The England midfielder showed a cool head after Michael Essien's effort had taken two deflections.
United had made the breakthrough thanks to Cristiano Ronaldo's 42nd goal of the season, the Portugal winger heading in at the back post in the 26th minute.
After the break, Chelsea regrouped thanks to Essien's equaliser and began to strengthen their grip on the game.
The Blues almost took the lead in the 78th minute when a marvellous 25-yard curler from Drogba beat Van der Sar only to come back off the post.
In the process, Giggs made his 759th appearance for the club to overtake Sir Bobby Charlton's record as the final went into extra-time.
Salomon Kalou was thrown on by Avram Grant for the start and Chelsea again hit the frame of the goal in the first period as Lampard struck Van der Sar's crossbar with a snapshot inside the United area. Then just before halftime, Giggs was thwarted by a wonderful peice of defending by John Terry.
Then came Drogba's stunning dismissal after an ugly all-in scuffle between the two sets of players. The Ivorian was shown a straight red card by ref Lubos Michels for slapping Nemanja Vidic in the face. The Serb didn't flinch, but that didn't stop Drogba seeking more confrontation with the United defender as he was ferried from the pitch.
The stats told the story for Chelsea, with the Blues having hit the target 24 times over the 120 minutes compared to United's 12.